LDA at Work

Falko AueRisk Analytics & Instruments1, Risk and Capital Management,Deutsche Bank AG, Taunusanlage 12, 60325 Frankfurt, Germany

Michael KalkbrenerRisk Analytics & Instruments, Risk and Capital Management,Deutsche Bank AG, Taunusanlage 12, 60325 Frankfurt, Germany

February 2007

Abstract

The Advanced Measurement Approach in the Basel II Accord permits anunprecedented amount of flexibility in the methodology used to assess OR cap-ital requirements. In this paper, we present the capital model developed atDeutsche Bank and implemented in its official EC process. The model followsthe Loss Distribution Approach. Our presentation focuses on the main quan-titative components, i.e. use of loss data and scenarios, frequency and severitymodelling, dependence concepts, risk mitigation, and capital calculation andallocation. We conclude with a section on the analysis and validation of LDAmodels.

Keywords: Loss Distribution Approach, frequency distribution, severity distribu-tion, Extreme Value Theory, copula, insurance, Monte Carlo, Economic Capital,model validation

1Deutsche Bank’s LDA model has been developed by the AMA Project Task Force, a collabora-tion of Operational Risk Management, Risk Analytics and Instruments, and Risk Controlling.

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Contents

1 Introduction 3

2 Survey of the LDA model implemented at Deutsche Bank 4

3 Loss data and scenarios 73.1 Importance of loss data . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Data sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Data classification and specification of BL/ET matrix . . . . . . . . 83.4 Use of data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.4.1 Use of internal loss data . . . . . . . . . . . . . . . . . . . . . 103.4.2 Incorporation of external loss data . . . . . . . . . . . . . . . 113.4.3 Incorporation of scenario analysis . . . . . . . . . . . . . . . . 11

4 Weighting of loss data and scenarios 124.1 Split losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Old losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.3 Scaling of external data and scenarios . . . . . . . . . . . . . . . . . 13

4.3.1 Characteristics of external data . . . . . . . . . . . . . . . . . 134.3.2 Scaling algorithms . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Frequency distributions 155.1 Data requirements for specifying frequency distributions . . . . . . . 155.2 Calibration algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6 Severity distributions 186.1 Complexity of severity modelling . . . . . . . . . . . . . . . . . . . . 186.2 Modelling decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6.2.1 Availability of data . . . . . . . . . . . . . . . . . . . . . . . . 196.2.2 Characteristics of data . . . . . . . . . . . . . . . . . . . . . . 206.2.3 Summary of modelling decisions . . . . . . . . . . . . . . . . 23

6.3 Specification of severity distributions . . . . . . . . . . . . . . . . . . 236.3.1 Building piecewise-defined distributions . . . . . . . . . . . . 236.3.2 Calibration of empirical distribution functions . . . . . . . . . 246.3.3 Calibration of parametric tail . . . . . . . . . . . . . . . . . . 24

7 Dependence 267.1 Types of dependence in LDA models . . . . . . . . . . . . . . . . . . 267.2 Modelling frequency correlations . . . . . . . . . . . . . . . . . . . . 30

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8 Risk mitigation 318.1 Insurance models in the Loss Distribution Approach . . . . . . . . . 318.2 Modelling insurance contracts . . . . . . . . . . . . . . . . . . . . . . 328.3 Mapping OR event types to insurance policies . . . . . . . . . . . . . 32

9 Calculation of Economic Capital and capital allocation 339.1 Risk measures and allocation techniques . . . . . . . . . . . . . . . . 339.2 Simulation of the aggregate loss distribution . . . . . . . . . . . . . . 359.3 Monte Carlo estimates for Economic Capital . . . . . . . . . . . . . 36

10 Incorporation of business environment and internal control factors 37

11 Analysis and validation of LDA models 3811.1 Model characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

11.1.1 Definition of a general class of LDA models . . . . . . . . . . 3911.1.2 Variance analysis . . . . . . . . . . . . . . . . . . . . . . . . . 3911.1.3 Loss distributions for heavy-tailed severities . . . . . . . . . . 41

11.2 Sensitivity analysis of LDA models . . . . . . . . . . . . . . . . . . . 4211.2.1 Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.2.2 Severities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.2.3 Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.2.4 Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

11.3 Impact analysis of stress scenarios . . . . . . . . . . . . . . . . . . . 4411.4 Backtesting and benchmarking . . . . . . . . . . . . . . . . . . . . . 44

12 References 47

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1 Introduction

A key demand on a bank’s Economic Capital methodology is to ensure that Eco-nomic Capital covers all material sources of risk. This requirement is a preconditionfor providing reliable risk estimates for capital management and risk-based perfor-mance measurement. Since operational losses are an important source of risk thequantification of operational risk has to be part of the calculation of a bank’s Eco-nomic Capital. A strong additional incentive for the development of a quantitativeOR methodology has been provided by the inclusion of operation risk into the Regu-latory Capital requirements under Pillar I of the Basel II Accord (Basel Committeeon Banking Supervision, 2006).

The Basel II Accord introduces three approaches to the quantification of opera-tion risk. The most sophisticated option is the Advanced Measurement Approach.It requires the calculation of a capital measure to the 99.9%-ile confidence levelover a one-year holding period.2 The Advanced Measurement Approach permits anunprecedented amount of flexibility in the methodology used to assess OR capitalrequirements, albeit within the context of strict qualifying criteria. This flexibilitysparked an intense discussion in the finance industry. Many quantitative and qualita-tive techniques for measuring operational risk have been proposed, most prominentlydifferent variants of the Loss Distribution Approach and techniques based on sce-narios and risk indicators. In our opinion, the natural way to meet the soundnessstandards for Economic and Regulatory Capital is by explicitly modelling the ORloss distribution of the bank over a one-year period. In this sense, AMA modelsnaturally follow the Loss Distribution Approach, differing only in how the loss dis-tribution is modelled.

The application of the LDA to the quantification of operational risk is a difficulttask. This is not only due to the ambitious soundness standards for risk capital butalso to problems related to operational risk data and the definition of operationalrisk exposure, more precisely

1. the shortage of relevant operational risk data,

2. the context dependent nature of operational risk data, and

3. the current lack of a strongly risk sensitive exposure measure in operationalrisk modelling (cf market and credit risk).

The main objective of an LDA model is to provide realistic risk estimates for thebank and its business units based on loss distributions that accurately reflect theunderlying data. Additionally, in order to support risk and capital management,the model has to be risk sensitive as well as sufficiently robust. It is a challenging

2Many banks derive Economic Capital estimates from even higher quantiles. For example, the99.98% quantile is used at Deutsche Bank.

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practical problem to find the right balance between these potentially conflicting goals.Finally, the model will only be accepted and implemented in the official processes ofa bank if it is transparent and produces explainable results.

In this paper, we present the LDA model developed and implemented at DeutscheBank. It is used for the quarterly calculation of OR Economic Capital since thesecond quarter of 2005. Subject to approval by regulatory authorities, the modelwill also be used for calculating Regulatory Capital.

The details of an LDA model are usually tailored to the specific requirements andlimitations of a bank, e.g. the availability of data has an impact on the granularityof the model, the weights given to the different data sources, etc. However, the basicstructure of LDA models as well as the fundamental modelling issues are rathersimilar across different banks. We therefore hope that the presentation of an LDAmodel that has been designed according to the Basel II guidelines and is part of thebank’s official EC process is regarded as an interesting contribution to the currentdebate.

Section 2 outlines the Loss Distribution Approach implemented at Deutsche Bankand provides a summary of this document. Our presentation focuses on the quanti-tative aspects of the model and their validation. Qualitative aspects like generationof scenarios or development of a program for key risk indicators are beyond the scopeof this paper.

2 Survey of the LDA model implemented at DeutscheBank

Figure 1 provides the flowchart of the model. Each of the components will be dis-cussed in the following sections.

The fundamental premise underlying LDA is that each firm’s operational lossesare a reflection of its underlying operational risk exposure (see subsection 3.1). Webelieve that loss data is the most objective risk indicator currently available. How-ever, even with perfect data collection processes, there will be some areas of thebusiness that will never generate sufficient internal data to permit a comprehensiveunderstanding of the risk profile. This is the reason why internal data is supple-mented by external data and generated scenarios: Deutsche Bank is a member ofThe Operational Riskdata eXchange Association, it has purchased a commercialloss database and has set up a scenario generation process. More information on thedifferent data sources is provided in subsection 3.2.

The first step to generating meaningful loss distributions, is to organize loss datainto categories of losses and business activities, which share the same basic risk profileor behaviour patterns. In subsection 3.3, we present the business line/event typematrix used in the model and discuss various criteria for merging cells. Subsection3.4 focuses on the incorporation of external loss data and scenarios analysis.

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Figure 1: Flowchart of LDA model.

In general, all data points are regarded as a sample from an underlying distribu-tion and therefore receive the same weight or probability in the statistical analysis.However, there are a number of exceptions: split losses, i.e. losses that are assignedto more than one business line, old losses, external losses in the commercial loss database and scenarios. Section 4 presents algorithms for adjusting the weights of thesedata points.

Whereas sections 3 and 4 deal with the data sources that are used in the modellingprocess, sections 5 and 6 are devoted to the specification of loss distributions. Moreprecisely, LDA involves modelling a loss distribution in each cell of the BL/ETmatrix. The specification of these loss distributions follows an actuarial approach:separate distributions for event frequency and severity are derived from loss data andthen combined by Monte Carlo simulation. In section 5, techniques are presentedfor calibrating frequency distributions and selecting the distribution that best fitsthe observed data.

OR capital requirements are mainly driven by individual high losses. Severitydistributions specify the loss size and are therefore the most important component

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in quantitative OR models. Severity modelling is a difficult problem. In particular,tails of severity distributions are difficult to estimate due to the inherent scarcityof low frequency, high impact operational loss events. The methodology applied inDB’s LDA model combines empirical distributions and parametric tail distributionswhich are derived with the Peaks-Over-Threshold method, a technique from ExtremeValue Theory (EVT). The severity model is presented in section 6.

The overall capital charge for the firm is calculated by aggregating the loss dis-tributions generated in the above fashion, ideally in a way that recognizes the risk-reducing impact of less than full correlation between the risks in each of the eventtype/business line combinations. In section 7, the most general mathematical con-cept for modelling dependence, so-called copulas, are applied to this aggregationproblem. More precisely, the frequency distributions in the individual cells of theBL/ET matrix are correlated through a Gaussian copula in order to replicate ob-served correlations in the loss data.

A realistic quantification of operational risk has to take the risk reducing effectof insurance into account. Compared to other methodologies a bottom-up LDA hasthe benefit of allowing a fairly accurate modelling of insurance cover. Transferringrisk to an insurer through insurance products alters the aggregate loss distributionby reducing the severity of losses that exceed the policy deductible amount. Thefrequency of loss is unaffected by insurance. More precisely, when frequency andseverity distributions are combined through simulation, each individual loss pointcan be compared to the specific insurance policies purchased by the bank and thecorresponding policy limits and deductibles. As a consequence, an insurance modelin the Loss Distribution Approach consists of two main components: a quantitativemodel of the individual insurance policies and a mapping from the OR event typesto the insurance policies. Both components are specified in section 8.

Section 9 focuses on the simulation of the aggregate loss distribution (includinginsurance) at Group level and on the calculation of Economic Capital and capitalallocation. Risk measures are based on a one-year time horizon. At Deutsche Bank,Economic Capital for operational risk (before qualitative adjustments) is defined asthe 99.98% quantile minus the Expected Loss. Expected Shortfall contributions areused for allocating capital to business lines, i.e. the contribution of a business lineto the tail of the aggregate loss distribution. For business units at lower hierarchylevels that do not permit the specification of separate loss distributions the capitalallocation is based on risk indicators instead.

Apart from generated loss scenarios LDA models mainly rely on loss data and areinherently backward looking. It is therefore important to incorporate a componentthat reflects changes in the business and control environment in a timely manner. InDB’s LDA model, qualitative adjustments are applied to the contributory capital ofbusiness lines. The direct adjustment of EC reduces the complexity of the model andimproves its transparency. However, it is difficult to justify with statistical means.Details of the incorporation of qualitative adjustments are given in section 10.

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The final section of this paper deals with model analysis and validation. Wepresent a sensitivity analysis of the model components frequencies, severities, depen-dence structure and insurance. The analysis uses basic properties of LDA modelsand is therefore not limited to the model implemented at Deutsche Bank. We brieflydeal with the impact analysis of stress scenarios and outline the inherent problemswith the application of backtesting techniques to OR models. However, the mainfocus of section 11 is on an approach for benchmarking quantiles in the tail of theaggregate loss distribution of the LDA model against individual data points fromthe underlying set of internal and external losses.

3 Loss data and scenarios

3.1 Importance of loss data

We believe that loss data is the most objective risk indicator currently available,which is also reflective of the unique risk profile of each financial institution. Lossdata should therefore be the foundation of an Advanced Measurement Approachbased on loss distributions (ITWG, 2003). This is one of the main reasons for un-dertaking OR loss data collection. It is not just to meet regulatory requirements,but also to develop one of the most important sources of operational risk manage-ment information. We acknowledge that internal loss data also has some inherentweaknesses as a foundation for risk exposure measurement, including:

1. Loss data is a ”backward-looking” measure, which means it will not immedi-ately capture changes to the risk and control environment.

2. Loss data is not available in sufficient quantities in any financial institution topermit a reasonable assessment of exposure, particularly in terms of assessingthe risk of extreme losses.

These weaknesses can be addressed in a variety of ways, including the use of sta-tistical modelling techniques, as well as the integration of the other AMA elements,i.e. external data, scenario analysis and factors reflective of the external risk andinternal control environments, all of which are discussed in the next sections of thisdocument.

3.2 Data sources

The following data sources are used in DB’s LDA model:

• Internal loss data: Deutsche Bank started the collection of loss data in 1999.A loss history of more than five years is now available for all business lines inthe bank.

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• Consortium data: loss data from The Operational Riskdata eXchange Associ-ation (ORX).

• Commercial loss data base: data from OpVantage, a subsidiary of Fitch Risk.

• Generated scenarios: specified by experts in divisions, control & support func-tions and regions.

The process of selecting, enhancing and approving the loss data from all sourcesand finally feeding it into the LDA model is named Relevant Loss Data Process inDeutsche Bank. We will not provide details but list a few principles:

• As long as data is considered relevant according to defined criteria for businessactivity (firm type, product and region), it will be included in the capital calcu-lations, no matter when the loss occurred (see section 4.2 for adjustments madeto old losses). This ensures that the largest possible meaningful population ofloss data is used, thus increasing the stability of the capital calculation.

• There is no adjustment to the size of the loss amount (scaling) in any datasource except for inflation adjustment. However, the weights of data pointsfrom the public data source are adjusted as outlined in section 4.3.

• Gross losses after direct recoveries are used for capital purposes. Insurancerecoveries are not subtracted at this stage because they are modelled separately.

• All losses are assigned to the current divisional structure. External data sourcesuse different business lines and have to be mapped to the internal structure.If possible a 1:1 mapping is performed. However, the framework also allowsmapping of one external business line to several internal business lines. Inthis case the weight of external data points is adjusted in order to reflect themapping percentage (compare section 4.1).

• Boundary events are excluded from OR capital calculations, e.g. Business Risk,Credit Risk, Market Risk, Timing Losses.

• FX conversion into EUR generally takes place the date the event was booked.In order to report all cash flows of an event with multiple cash flows consis-tently, the FX rate of the first booking date is used.

• Procedures for avoiding double counting between data sources are in place.

3.3 Data classification and specification of BL/ET matrix

The first step to generating meaningful loss distributions, is to organize loss data intocategories of losses and business activities, which share the same basic risk profileor behaviour patterns. For instance, we expect that fraud losses in Retail Banking

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will share a unique loss distribution, which may be quite different from employeeclaims in the Investment Banking business. If all losses are lumped together it maybe difficult to discern a pattern, whereas if they are separated it becomes easier todescribe a unique risk profile and be confident that it is a realistic picture of potentialexposure. The Basel Committee has specified a standard matrix of risk types andbusiness lines to facilitate data collection and validation across the various AMAapproaches (Basel Committee, 2002a). Firms using LDA are required to map theirloss data to the standard matrix, and prove that they have accounted for losses in allaspects of their operations, without being further restricted as to how they actuallymodel the data. In other words, any given firm may choose to collapse or expandthe cells in the matrix for purposes of building a specific loss distribution. DeutscheBank’s BL/ET matrix is specified according to

• the internal business lines represented in the Executive Committee of DeutscheBank and

• Level 1 of the Basel II event type classification.3

The decision whether a specific cell is separately modelled or combined with othercells depends on several factors. The following criteria have been identified:

• comparable loss profile

• same insurance type

• same management responsibilities

Other important aspects are data availability and the relative importance of cells.Based on these criteria the seven Basel II event types have been merged into fiveevent types:

Fraud Internal FraudExternal Fraud

Infrastructure Damage to Physical AssetsBusiness Disruption, System Failures

Clients, Products, Business PracticesExecution, Delivery, Process ManagementEmployment Practices, Workplace Safety

Fraud and Clients, Products, Business Practices and Execution, Delivery, ProcessManagement are the dominant risk types in terms of the number of losses as well asthe width of the aggregated loss distribution. As a consequence, these event typesare modelled separately by business line whereas Infrastructure and Employment aremodelled across business lines. This results in the BL/ET matrix in Table 1.

There exist loss events that cannot be assigned to a single cell but3We refer to Samad-Khan (2002) for a critical assessment of the Basel II event type classification.

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Table 1: BL/ET matrix.

• affect either the entire Group (Group losses)

• or more than one business line (split losses).

The cells 7, 15 and 22 are used for modelling Group losses. The modelling andallocation techniques applied in these Group cells are identical to the techniques inthe divisional cells.

Some losses consist of several components that are assigned to different businesslines but have the same underlying cause. For example, consider a penalty of 100mthat has been split between business line A (70m) and business line B (30m). If thetwo components of 70m and 30m are modelled separately in the respective divisionalcells, their dependence would not be appropriately reflected in the model. This wouldinevitably lead to an underestimation of the risk at Group level. This problemis avoided by aggregating the components of a split loss and assigning the totalamount to each of the cells involved. However, the weights (or probabilities) of thecomponents of a split loss are reduced accordingly: in the above example, the totalamount of 100m is assigned to both business lines but the weight of this event isonly 70% for business line A and 30% for business line B. We refer to section 4 formore information on adjustments to the weights of loss events and scenarios.

3.4 Use of data

3.4.1 Use of internal loss data

Deutsche Bank’s internal losses are the most important data source in its model.Internal loss data is used for

1. modelling frequency distributions,

2. modelling severity distributions (together with external losses and scenarios),

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3. analyzing the dependence structure of the model and calibrating frequencycorrelations.

3.4.2 Incorporation of external loss data

It seems to be generally accepted in the finance industry that internal loss data aloneis not sufficient for obtaining a comprehensive understanding of the risk profile of abank. This is the reason why additional data sources have to be used, in particularexternal losses (Basel Committee on Banking Supervision, 2006).

There are many ways to incorporate external data into the calculation of oper-ational risk capital. External data can be used to supplement an internal loss dataset, to modify parameters derived from the internal loss data, and to improve thequality and credibility of scenarios. External data can also be used to validate theresults obtained from internal data or for benchmarking.

In DB’s LDA model, external data is used as additional data source for modellingtails of severity distributions. The obvious reason is that extreme loss events at eachbank are so rare that no reliable tail distribution can be constructed from internaldata only. We are well aware that external losses do not reflect Deutsche Bank’srisk profile as accurately as internal events but we still believe that they significantlyimprove the quality of the model.4 In the words of Charles Babbage (1791-1871):“Errors using inadequate data are much less than those using no data at all.”

3.4.3 Incorporation of scenario analysis

Scenario analysis is another important source of information. In this paper, welimit the discussion to the application of scenarios in DB’s LDA model and referto Scenario-Based AMA Working Group (2003), ITWG Scenario Analysis WorkingGroup (2003), Anders and van den Brink (2004), Scandizzo (2006) and Alderweireldet al. (2006) for a more thorough discussion of scenario analysis including the designand implementation of a scenario generation process.

From a quantitative perspective, scenario analysis can be applied in several ways(ITWG Scenario Analysis Working Group, 2003):

• To provide data points for supplementing loss data, in particular for tail events

• To generate loss distributions from scenarios that can be combined with lossdistributions from loss data

• To provide a basis for adjusting frequency and severity parameters derivedfrom loss data

• To stress loss distributions derived from loss data4The direct application of external loss data is a controversial issue. See, for example, Alvarez

(2006) for a divergent opinion.

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The main application of generated scenarios in DB’s LDA model is to supplementloss data. More precisely, the objective of scenario analysis is to capture high impactevents that are not already reflected in internal or external loss data. Starting pointfor the integration of scenarios is the set of relevant losses in OpVantage. Theselosses have been selected in the Relevant Loss Data Process and can therefore beregarded as one-event scenarios. In the next step, scenarios are generated as deemednecessary to fill in potential severe losses not yet experienced in the past. Eachscenario contains a description and an estimate of the loss amount. The processfor the generation of scenario descriptions and severities is driven by experts indivisions, control & support functions and regions and is followed by a validationprocess. The scenario data points are combined with the relevant OpVantage dataand receive the same treatment, i.e. scaling of probabilities of individual data points.The combined data set of relevant OpVantage losses and generated scenarios is animportant element in the calibration of the tails of severity distributions.

4 Weighting of loss data and scenarios

Loss data and scenarios are used for calibrating frequency and severity distributions.In general, all data points are regarded as a sample from an underlying distribu-tion and therefore receive the same weight or probability in the statistical analysis.However, there are three exceptions:

1. split losses

2. old losses

3. external losses in the commercial loss data base and scenarios

4.1 Split losses

Split losses are loss events that cannot be assigned to a single cell but affect morethan one business line. The treatment of split losses has already been discussed insection 3.3: the total amount of a split loss is assigned to each business line affected,the weight being set to the ratio of the partial amount of the respective business linedivided by the aggregated loss amount. Note that the sum of the weights of a splitloss equals one.

4.2 Old losses

Since the risk profile of a bank changes over time, old losses will be less representative.The impact of a given loss should therefore be reduced over an appropriate timeperiod. In DB’s LDA model, the phasing-out of old losses is implemented in thefollowing way:

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• For frequency calibration, only internal losses are used that occurred in thelast five years.

• For severity calibration and scaling of OpVantage losses, a weighting by timeis introduced. Losses that occurred within the last 5 years receive full weight.The weight of older losses is linearly decreased from one to zero over a periodof 20 years.

4.3 Scaling of external data and scenarios

4.3.1 Characteristics of external data

External loss data is inherently biased. The following problems are typically associ-ated with external data:

• Scale bias - Scalability refers to the fact that operational risk is dependent onthe size of the bank, i.e. the scale of operations. A bigger bank is exposedto more opportunity for operational failures and therefore to a higher level ofoperational risk. The actual relationship between the size of the institutionand the frequency and severity of losses is dependent on the measure of sizeand may be stronger or weaker depending on the particular operational riskcategory.

• Truncation bias - Banks collect data above certain thresholds. It is generallynot possible to guarantee that these thresholds are uniform.

• Data capture bias - Data is usually captured with a systematic bias. This prob-lem is particularly pronounced with publicly available data. More precisely, onewould expect a positive relationship to exist between the loss amount and theprobability that the loss is reported. If this relationship does exist, then thedata is not a random sample from the population of all operational losses,but instead is a biased sample containing a disproportionate number of verylarge losses. Standard statistical inferences based on such samples can yieldbiased parameter estimates. In the present case, the disproportionate numberof large losses could lead to an estimate that overstates a bank’s exposure tooperational risk (see de Fontnouvelle et al. (2003)).

4.3.2 Scaling algorithms

In the current version of the model, external loss data is not scaled with respect tosize. The reason is that no significant relationship between the size of a bank and theseverity of its losses has been found in a regression analysis of OpVantage data doneat Deutsche Bank (compare to Shih et al. (2000)). This result is also supported byan analysis of internal loss data categorized according to business lines and regions.

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General techniques for removing the truncation bias can be found in Baud et al.(2002) and Chernobai et al. (2006). In the frequency and severity models presentedin this paper, the truncation bias does not pose a problem.

We scale publicly available loss data in order to remove the data collection bias.5

The basic idea is to adjust the probabilities (and not the size) of the external lossevents in order to reflect the unbiased loss profile, i.e. increase the probability ofsmall losses and decrease the probability of large losses.6 The crucial assumptionin our approach is that ORX data and (unbiased) OpVantage data have the samerisk profile, i.e. both reflect the generic OR profile of the finance industry. ORXdata is assumed to be unbiased. As a consequence, the probabilities of the publicloss events are adjusted in such a way that the OpVantage loss profile (after scaling)reflects the ORX profile. The scaling is performed at Group level, i.e. it is based onall OpVantage losses, scenarios and ORX events (including losses in DB) above 1m.The same scaling factors are applied across all business lines and event types.

The mathematical formalization of the scaling technique is based on stochasticthresholds. More precisely, following Baud et al. (2002) and de Fontnouvelle et al.(2003) we extract the underlying (unbiased) loss distribution by using a model inwhich the truncation point for each loss (i.e., the value below which the loss is notreported) is modelled as an unobservable random variable. As in de Fontnouvelle etal. (2003) we apply the model to log losses and assume that the distribution of thethreshold is logistic. However, our model does not require any assumptions on thedistribution of the losses.7 We will now describe the model in more detail.

Let X1, . . . , Xm be independent samples of a random variable X. The variableX represents the “true” loss distribution and is identified with ORX data. LetY1, . . . , Yn be independent samples of the conditional random variable Y := X|H ≤X, where H is another random variable (independent of X) representing the stochas-tic threshold. We assume that the distribution function Fθ(x) of H belongs to aknown distribution class with parameters θ = (θ1, . . . , θr). The variable Y is iden-tified with OpVantage data. The objective of scaling is to determine the thresholdparameters θ = (θ1, . . . , θr) from the data sets {X1, . . . , Xm} and {Y1, . . . , Yn}. Thecriterion for parameter calibration is to minimize

k∑i=1

(P(H ≤ X ≤ Si |H ≤ X)− P(Y ≤ Si))2 ,

where S1, . . . , Sk are positive real numbers, i.e. severities. The probabilities P(Y ≤Si) are derived from the samples Y1, . . . , Yn. The probabilities P(H ≤ X ≤ Si |H ≤

5More precisely, the data set consisting of relevant OpVantage losses and generated scenarios(see section 3.4.3) is scaled.

6Since external data is not used for modelling frequencies the scaling methodology affects severitydistributions only.

7In contrast, de Fontnouvelle et al. (2003) assume that the distribution of excesses of logarithmsof reported losses converges to the exponential distribution which is a special case of the GPD.

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X) are calculated asLi∑

j=1

Fθ(Xj)/m∑

j=1

Fθ(Xj),

where Li is the highest index such that XLi ≤ Si and w.l.o.g. X1 ≤ . . . ≤ Xm.The following parameter setting is used in the current model: in order to specify

X1, . . . , Xm, all ORX and internal losses above 1m EUR are selected. The samplesY1, . . . , Yn are the relevant OpVantage losses and scenarios. The threshold is assumedto follow a loglogistic distribution, i.e. it has the distribution function

Fµ,β(x) =1

1 + e− log(x)−µ

β

,

which is equivalent to applying a logistic threshold to log losses. This distributionhas been chosen because it provides an excellent fit to our data.

Figure 2 displays the impact of scaling on the OpVantage profile by means of twoQQ-plots: the unscaled OpVantage date is much heavier than the consortium datawhereas the profiles match rather well after scaling.

5 Frequency distributions

The standard LDA uses actuarial techniques to model the behaviour of a bank’soperational losses through frequency and severity estimation, i.e. the loss distribu-tion in each cell of the BL/ET matrix is specified by separate distributions for eventfrequency and severity. Frequency refers to the number of events that occur withina given time period. Although any distribution on the set of non-negative integerscan be chosen as frequency distribution, the following three distribution families areused most frequently in LDA models: the Poisson distribution, the negative binomialdistribution, or the binomial distribution (see Johnson et al. (1993) or Klugman etal. (2004) for more information).

5.1 Data requirements for specifying frequency distributions

In DB’s LDA model, the specification of frequency distributions is entirely based oninternal loss data (in contrast to Frachot and Roncalli (2002) who suggest to useinternal and external frequency data8). The main reasons for using only internaldata are:

• Internal loss data reflects DB’s loss profile most accurately.8If external loss data is used for frequency calibration the external frequencies have to be scaled

based on the relationship between the size of operations and frequency.

16

Figure 2: QQ-plots showing the log quantiles of internal and consortium losses above1m on the x-axis and the log quantiles of public loss data and scenarios on the y-axis.

• It is difficult to ensure completeness of loss data from other financial institu-tions. However, data completeness is essential for frequency calibration.

• Data requirements are lower for calibrating frequency distributions than forcalibrating severity distributions (in particular, if Poisson distributions areused).

For calibrating frequency distributions, time series of internal frequency data areused in each cell. Frequency data is separated into monthly buckets in order toensure that the number of data points is sufficient for a statistical analysis.

5.2 Calibration algorithms

We have implemented calibration and simulation algorithms for Poisson, binomialand negative binomial distributions. In order to determine the appropriate distribu-tion class for a particular cell we apply three different techniques to the correspondingtime series.

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• The dispersion of the time series is analyzed by comparing its mean andvariance.9 If the time series is equidispersed a Poisson distribution is used.In case of overdispersion (underdispersion) the frequency distribution is mod-elled as a negative binomial distribution (binomial distribution). Since it isnot obvious which mean/variance combinations should be considered equidis-persed the results of the dispersion analysis are compared with the followinggoodness-of-fit tests.

• We estimate the parameters of a Poisson distribution and a negative binomialdistribution by matching the first two moments10. Then a goodness-of-fit test(to be more precise, a χ2-test) is used to analyze the hypothesis of a Pois-son and a negative binomial distribution respectively. The idea of the χ2-testis based on comparisons between the empirically measured frequencies andthe theoretically expected frequencies. More precisely, the frequencies of theobserved data are aggregated on chosen intervals and compared to the theoret-ically expected frequencies. The sum over the (weighted) squared differencesfollows approximately a specific χ2-distribution. This can be used to calculatefor a given level (e.g. α = 0.05) a theoretical “error”. If the observed error isgreater than the theoretical error, the hypothesis is rejected. The level α canbe understood as the probability to reject a true hypothesis. In order to avoidthe (subjective) choice of the level α, one can calculate a so called p-value,which is equal to the smallest α–value at which the hypothesis can be rejected,based on the observed data.

• Another test that we perform in this context analyzes the interarrival timeof losses. If the data has been drawn from an independent random process(e.g. Poisson Process), the interarrival times of the losses follow an exponen-tial distribution. The interarrival times are calculated over a particular timehorizon and fitted by an exponential density, whose parameter is estimatedwith a ML-estimator. A χ2-test is used to assess the quality of the fit.

Based on these tests we have developed an algorithm for selecting a Poisson ornegative binomial distribution. Furthermore, we have analyzed the impact of differ-ent distribution assumptions on Economic Capital. It turned out that it is almostirrelevant for EC at Group and cell level whether Poisson or negative binomial dis-tributions are used. This result agrees with the theoretical analysis in section 11(see also Bocker and Kluppelberg (2005) and De Koker (2006)): in LDA modelsapplied to OR data, the choice of severity distributions usually has a much moresevere impact on capital than the choice of frequency distributions. It has therefore

9A distribution is called equidispersed (overdispersed; underdispersed) if the variance equals(exceeds; is lower than) the mean.

10Since no underdispersed time series were observed binomial distributions are not considered.

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been decided to exclusively use Poisson distributions in the official capital calcula-tions at Deutsche Bank. This decision reduces the complexity of the model since nostatistical tests or decisions rules for frequency distributions are required.

6 Severity distributions

6.1 Complexity of severity modelling

OR capital requirements are mainly driven by individual high losses. Severity dis-tributions specify the loss size and are therefore the most important component inquantitative OR models.

Severity modelling is a difficult task. One reason is the lack of data. Internal lossdata covering the last 5 to 7 years is not sufficient for calibrating tails of severitydistributions. It is obvious that additional data sources like external loss data andscenarios are needed to improve the reliability of the model. However, inclusion ofthis type of information immediately leads to additional problems, e.g. scaling ofexternal loss data, combining data from different sources, etc.

Even if all available data sources are used it is necessary to extrapolate beyondthe highest relevant losses in the data base. The standard technique is to fit aparametric distribution to the data and to assume that its parametric shape alsoprovides a realistic model for potential losses beyond the current loss experience.The choice of the parametric distribution family is a non-trivial task and usually hasa significant impact on model results (compare, for example, to Dutta and Perry(2006) or Mignola and Ugoccioni (2006)).

Our experience with internal and external loss data has shown that in many cellsof the BL/ET matrix the body and tail of the severity distribution have differentcharacteristics. As a consequence, we have not been able to identify parametricdistribution families in these cells that provide acceptable fits to the loss data acrossthe entire range. A natural remedy is to use different distribution assumptions forthe body and the tail of these severity distributions. However, this strategy addsanother layer of complexity to the severity model.

In summary, severity modelling comprises a number of difficult modelling ques-tions including:

Treatment of internal and external loss data and scenarios

• How much weight is given to different data sources?

• How to combine internal and external data and scenarios?

Range of distribution

• One distribution for the entire severity range or different distributions for small,medium and high losses?

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Choice of distribution family

• Two-parametric distributions like lognormal and GPD, more flexible paramet-ric distribution families, i.e. three- or four-parametric, or even empirical dis-tributions?

• One distribution family for all cells or selection of “best” distribution based onquality of fit?11

The main objective is to specify a realistic severity profile: severity distributionsshould provide a good fit to the available loss data over the entire range, in particularin the tail sections. However, fitting the data is not the only objective of severitymodelling. Since the severity model is the key driver of OR capital requirements thesensitivity of severity distributions to changes in the input data (losses and scenarios)is of particular importance. While a capital model used for risk measurement andsteering has to be risk sensitive wild swings in capital estimates are not acceptable forcapital planning and performance measurement. It is a difficult task to find the rightbalance between these potentially conflicting goals. Another import requirement fora capital model used in practice is that its structure and results should be explainableto non-quants. Again, this is quite challenging for severity modelling: the severitymodel has to be sophisticated enough to capture complex severity profiles but it hasto be kept as simple and transparent as possible in order to increase acceptance bybusiness and management.

6.2 Modelling decisions

The availability of internal loss data differs significantly across financial institutionsand there is no consensus about the application of external losses to severity mod-elling. It is therefore not surprising that there has not yet emerged a standardseverity model that is generally accepted in the industry. In this subsection, we dis-cuss the availability and characteristics of internal and external loss data at DeutscheBank and present the basic structure of the severity model derived from the data.

6.2.1 Availability of data

In a typical cell of the BL/ET matrix, there are sufficient internal data points tospecify a reliable severity profile between 10k (the internal data collection thresh-old) and 1m. However, the number of internal losses above 1m is rather limited.We therefore use all cell-specific external losses and scenarios as an additional datasource. However, even if all data points in a cell are combined we do not have suf-ficient information on the extreme tail of the severity distribution, say beyond 50m.

11A description of goodness-of-fit tests like Kolmogorov-Smirnov test, Anderson-Darling test, QQ-plots, etc and their application to OR data can be found e.g. in Chernobai et al. (2005), Moscadelli(2004) or Dutta and Perry (2006).

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This is the reason why a third data source is used: all internal losses, external lossesand scenarios (across all cells) above 50m. This choice is based on the underlyingassumption that the extreme tails of the severity distributions in the different cellshave something in common. Of course, this assumption is debatable. However, weconsider it the better option compared to extrapolating far beyond the highest lossthat has occurred in a particular cell.

6.2.2 Characteristics of data

It seems to be generally accepted in the finance industry that OR capital require-ments of large international banks are mainly driven by rare and extreme losses.This fact has a strong influence on the choice of the distribution families used formodelling severities in operational risk: it is quite natural to work with distributionsthat have been applied in insurance theory to model large claims. Many of thesedistributions belong to the class of subexponential distributions (see section 11 andEmbrechts et al. (1997) for more information). Examples are Pareto, Weibull (withτ < 1), Benktander-type-I and II, lognormal and loggamma distributions.12

We have experimented with a number of subexponential distributions, includingtruncated13 lognormal, Weibull and Pareto. When we fitted these distributions tothe internal and external data points we encountered two main problems:

1. In many cells of the BL/ET matrix, the body and tail of the severity distribu-tion have different characteristics. As a consequence, we have not been able toidentify parametric distribution families in these cells that provide acceptablefits to the loss data across the entire range. Typically, the calibrated distri-bution parameters were dominated by the large number of losses in the bodywhich resulted in a poor fit in the tail.

2. Calibration results were rather unstable, i.e. for rather different pairs of distri-bution parameters the value of the maximum-likelihood function was close toits maximum. In other words, these different parametrizations provided a fitof comparable quality to the existing data points. Even different distributionfamilies frequently resulted in a similar goodness-of-fit. In most cases, however,the calibrated distributions differed significantly in the extreme tails (compareto Mignola and Ugoccioni (2006)).

12In the literature, the calibration of various light and heavy-tailed distribution classes to oper-ational risk data is analyzed. de Fontnouvelle and Rosengren (2004) discuss the properties of themost common severity distributions and fit them to OR data. A similar study can be found inMoscadelli (2004). Dutta and Perry (2006) also examine a variety of standard distributions as wellas 4-parametric distributions like the g-and-h distributions and the Generalized Beta distributionof Second Kind. Alvarez (2006) suggests the 3-parametric lognormal-gamma mixture.

13The truncation point is 10k in order to reflect the internal data collection threshold. Chernobaiet al. (2006) analyse the errors in loss measures when fitting non-truncated distributions to truncateddata.

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A potential solution is to apply more flexible parametric distribution families,e.g. distributions with more than two parameters. However, even if the additionalflexibility improves the fit to the existing data points it seems doubtful whether thesedistributions provide a more reliable severity profile across the entire range.

Instead of applying high-parametric distribution families we have decided tomodel body and tail separately. Empirical distributions are used for modelling thebody of severity distributions. This approach offers the advantages that no choiceof a parametric distribution has to be made, the severity profile is reflected mostaccurately and high transparency is ensured.

For modelling severity tails, however, empirical distributions are clearly not suffi-cient. We combine empirical distributions with a parametric distribution in order toquantify the loss potential beyond the highest experienced losses. For the specifica-tion of the parametric distribution we have decided to apply Extreme Value Theory(EVT) or - more precisely - the Peaks-Over-Threshold method.

Extreme Value Theory is concerned with the analysis of rare and extreme eventsand therefore provides a natural framework for modelling OR losses. Most relevantfor the application in operational risk is a theorem in EVT saying that, for a certainclass of distributions, the generalized Pareto distribution (GPD) appears as limitingdistribution for the distribution of the excesses Xi − u, as the threshold u becomeslarge. Hence, this theorem provides guidance for selecting an appropriate distribu-tion family for modelling tails of severity distributions. Its algorithmic version isthe Peaks-Over-Threshold method. We refer to Embrechts et al. (1997) for an ex-cellent introduction to Extreme Value Theory and to Medova (2000), Cruz (2002),Embrechts et al. (2003), Moscadelli (2004) and Makarov (2006) for an applicationof EVT to OR data.

Unfortunately, the application of Extreme Value Theory in operational risk isnot straightforward. In the words of Chavez-Demoulin et al. (2005): “Applyingclassical EVT to operational loss data raises some difficult issues. The obstaclesare not really due to a technical justification of EVT, but more to the nature ofthe data”. Depending on the data set used, the papers cited in this section (see,moreover, Neslehova et al. (2006) and Mignola and Ugoccioni (2005)) come todifferent conclusions about the applicability of EVT to OR data. Our own experienceis summarized in the following paragraph.

Generalized Pareto distributions are specified by two parameters, the shape andthe scale parameter. Theory predicts that the calibrated shape parameter stabilizesas the threshold u becomes large and the distribution of excesses Xi−u converges toa GPD. It depends on the underlying loss data at which threshold the correspondingshape becomes constant. However, when we apply the Peaks-Over-Threshold methodto all losses and scenarios above 50m we do not observe stable shape parametersfor large thresholds. On the contrary, shape parameters tend to decrease whenthresholds are increased (compare to figure 3). This phenomenon is not necessarilycontradicting theory but may be caused by the lack of loss data: additional extreme

22

losses would enable us to increase thresholds further which might eventually leadto stable shape parameters. In the current setting, however, the decreasing shapesmake the modelling of tails more difficult.14 This problem will be addressed in moredetail in the following subsections.

Figure 3: Estimated shape parameter and number of excesses for various POTthresholds.

Despite this problem we consider the POT method the most reliable techniquefor modelling tails of severity distributions. Other distributions, e.g. truncatedlognormal and Weibull, used for modelling the extreme tail did not improve thequality of the fit and proved to be very unstable with respect to small deviations inthe loss data.

14Note that we would observe decreasing shape parameters if the size of individual OR lossescannot be infinite but is capped at a certain level: in this case the shapes would eventually convergeto a negative value.

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6.2.3 Summary of modelling decisions

In each cell, the severity distribution is piece-wise defined with three disjunctiveranges:

1. In each cell, we derive separate severity distributions

• from all internal losses >10k and• from all external losses and scenarios >1m respectively.

Another severity distribution is calibrated from all losses >50m (across allcells) that provides additional information on the tail.

2. Both cell-specific distributions are directly derived from the historic losses, i.e.are modelled as empirical distributions.

3. The Peaks-Over-Threshold method is used to calibrate the parametric distri-bution >50m.

4. Empirical and parametric distributions are combined to a piece-wise definedseverity distribution via a weighted sum applied to the cumulative distributionfunctions.

The following subsection provides details on the specification of the different distri-butions and the way they are combined.

6.3 Specification of severity distributions

6.3.1 Building piecewise-defined distributions

As outlined in section 6.2.3, different severity distributions are constructed at thethresholds 10k, 1m and 50m and are then aggregated to a piece-wise defined distri-bution with three disjunctive ranges. The cumulative distribution function F1 of thefirst range (10k - 1m) is based on cell specific, internal losses. The CDF F2 for thesecond range (1m - 50m) takes also cell specific, external data into account and thedistribution F3 for the tail (>50m) is derived from cell specific data and the data setconsisting of all losses (internal, ORX, scaled OpVantage and scenarios) above 50m.

With T1 = 1m and T2 = 50m, the combined severity distribution F is given by:

F (x) =

F 1(x) x < T1,

F 1(T1) · F 2(x) T1 ≤ x < T2,

F 1(T1) · F 2(T2) · F 3(x) T2 ≤ x,

where F (x) = 1− F (x) denotes the tail of F .

In detail:

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• F1 is given by the empirical distribution of cell specific internal data.

• F2 = wintFT1int + wextF

T1ext, with F T1

int being the truncated empirical distributionof cell specific internal data above T1 and F T1

ext the empirical distribution of cellspecific ORX, scaled OpVantage and scenario data above T1.

• F3 = wintFT2int + wextF

T2ext + wtailFtail. The empirical distributions of internal

and external data above T2 are taken into account in order to capture the cellspecific risk profile. Ftail is the parametric distribution calibrated on all lossesabove T2.

The setting of the weights has not been determined by statistical techniques but ismotivated by the following analyses and considerations:

• The weight of internal data increases with the number of internal data points:as internal data is considered most representative for the bank’s risk profile itreceives a high weight as long as sufficiently many internal losses are availableto ensure stability.

• The weight for the parametric tail function reflects the contribution of therespective event type to losses >50m.

• Impact analyses have been performed to investigate the volatility due to inclu-sion or exclusion of data points.

6.3.2 Calibration of empirical distribution functions

In principle, the derivation of empirical distributions from loss data is straightfor-ward. Note, however, that the data points in the loss sample receive different weights(compare section 4). Therefore the standard estimator for empirical distributions hasto be modified and reads

F T (x) =m∑

j=Li

w(Xj) · 1{Xj≤x}/m∑

j=Li

w(Xj),

where X1, . . . , Xm is the loss sample, Li is the lowest index such that XLi ≥ T andw.l.o.g. X1 ≤ . . . ≤ Xm. The weights w(X1), . . . , w(Xm) reflect the adjustments forthe split ratios, time decay and data collection bias.

6.3.3 Calibration of parametric tail

The calibration of the parametric tail is done in two steps. First scale and shapeparameters of GPDs are estimated at increasing excess thresholds u1, . . . , uK viathe Peaks-Over-Threshold method. As outlined in section 6.2.3 we observe that theshape parameters do not become constant but decrease with increasing thresholds.

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Thus, a modified GPD, which is a sequence of GPDs with varying scale and shapeparameters, is constructed in the second step. This approach is a modification of thestandard POT-method and provides the best and most stable fit to our data. Bothsteps are now described in detail.

The objective of the Peaks-Over-Threshold method is to fit a Generalized Paretodistribution to the excesses X1−u, . . . , Xn−u, where X1, . . . , Xn is the loss sample.For determining the scale and shape parameters we apply Maximum LikelihoodEstimation (MLE).15 The Maximum Likelihood Estimator calculates parameters ofthe distribution that maximize the “likelihood”. In our application, the likelihoodfunction is the density function of the Generalized Pareto distribution. Our losssample consists of internal losses, generated scenarios and data points from ORXand OpVantage. A number of techniques for combining loss data from differentsources have been proposed in recent years. Each of these techniques is based onspecific assumptions on the relationship between the severity distributions of thesesources.

The most straightforward approach assumes that all data sets follow the sameseverity distribution. Under this assumption direct mixing of data is possible, i.e.losses of all data sources are combined and used as input for the calibration of thejoint severity distribution.

To derive the parameters of a GPD from unbiased and biased data sources wefollow the methodology in Frachot and Roncalli (2002) and Baud et al. (2002). Itis based on the assumption that biased data, i.e. OpVantage and scenarios, havethe same distribution as unbiased data, i.e. internal and ORX, except that biaseddata are truncated below a stochastic threshold Hθ. The threshold is assumed tofollow a loglogistic distribution with parameters θ = (θ1, θ2). Its purpose is to scaleOpVantage to the generic OR profile of the finance industry represented by ORXand internal data (see subsection 4.3). Note, however, that for losses >50m thescaling function used to remove the data collection bias of the OpVantage data hasinsignificant influence and therefore does not affect the stability of the calibration.In this setting, MLE is applied to the log-likelihood

lθ(δ) =m∑

i=1

w(Xi) · ln(fδ(Xi − u)) +n∑

j=1

w(Yj) · ln(fθ,u,δ(Yj − u)),

where X1, . . . , Xm and Y1, . . . , Yn are series of internal/ORX and OpVantage/scenariodata above a threshold u in a given cell, fδ(x) is the density function of the underly-ing Generalized Pareto distribution with scale and shape parameters δ = (b, t) andfθ,u,δ(x) is the truncated density function of the severity distribution with stochastictruncation point Hθ and excess threshold u. The weights w(·) reflect the adjustmentsfor time decay and the special treatment of split losses as outlined in section 4.

15We have compared Maximum Likelihood Estimation to Hill Estimators (Embrechts et al., 1997)and have obtained consistent results.

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MLE is used to calculate the parameters δ = (b, t) of the severity distribution, theparameters θ = (θ1, θ2) have been calibrated in the scaling algorithm.16 Applyingthis methodology to the set of excess thresholds u1, . . . , uK yields a set of scale andshape parameters δ1 = (b1, t1), . . . , δK = (bK , tK).

In the second step we identify functions bβ(u) and tξ(u) with parameters β =(β1, . . . , βi) and ξ = (ξ1, . . . , ξj) that describe the behaviour of the scale and shapeparameters as functions of the threshold u. The precise form of bβ(u) and tξ(u)including the number of parameters i and j depends on the characteristics of theunderlying data. Analysis on several data cycles showed that shapes decrease in theorder of 1/u and scales increase in the order of ln(u). The parameters β and ξ areset such that the quadratic distance between estimated parameters δk = (bk, tk) andfunctions bβ(u) and tξ(u) taken at a selection of excess thresholds uk is minimized.The two functions bβ(u) and tξ(u) are then used to define a sequence of n GPDswith varying scale and shape parameters for excesses x over T2 = 50m.

An alternative approach to the piecewise specification of the parametric tail assequence of n GPDs is based on the corresponding limit distribution: for n → ∞,i.e. for infinite granularity, the tail distribution becomes

Gβ,ξ(x) := 1− limn→∞

n∏i=1

(1−G

bβ

(x(i−1)

n+T2

),tξ

(x(i−1)

n+T2

) (x

n

)).

This distribution does not depend on the parametric shape tξ(u) and has the form17

Gβ(x) = 1− exp(−

∫ T2+x

T2

1bβ(u)

du

). (1)

It can be calibrated from loss data using the standard maximum-likelihood estimator.Finally, it is interesting to note that if a linear scale function bβ(u) = β2 ·(u−T2)+β1

is used, the distribution (1) becomes a GPD with scale and shape parameters β1 andβ2.

7 Dependence

7.1 Types of dependence in LDA models

In a bottom-up Loss Distribution Approach, separate distributions for loss frequencyand severity are modelled at “business line/event type” level and then combined.

16For extensions or alternatives to Maximum-Likehood-Estimation we refer to de Fontnouvelleand Rosengren (2004) who adapt a regression-based EVT technique that corrects for small-samplebias in the tail parameter estimate proposed by Huisman et al. (2001). See also Peng and Welsh(2001) for other robust estimators.

17We are grateful to Andreas Zapp for pointing out this fact to us.

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More precisely, in each cell (k = 1, . . . ,m) of the BL/ET matrix the loss distributionis specified by

Lk =Nk∑i=1

Ski,

where Nk is the frequency distribution and Sk1, . . . , SkNkare random samples of the

severity distribution Sk. The loss distributions in the cells are then aggregated to aloss distribution at Group level that determines the Economic Capital of the bank.

In this basic model, dependencies between the occurrence of loss events and thesize of losses might occur in numerous ways:

Within a cell

• Dependence between the occurrence of loss events in a cell

• Dependence between the frequency distribution Nk and the severity distribu-tion Sk in a cell

• Dependence between the severity samples Sk1, . . . , SkNkin a cell

Between cells

• Dependence between the frequency distributions N1, . . . , Nm in different cells

• Dependence between the severity distributions S1, . . . , Sm in different cells

In this section, we analyze these types of dependence and discuss the consequencesfor modelling. Our analyses are based on time series of internal frequency andseverity data covering 5 years. In order to increase the number of data buckets weuse monthly intervals.

The statistical analysis of the dependence structure of OR losses does not onlyrequire information on the size of losses but also on the date of their occurrence.18

The quality of the results crucially depends on the quality of the underlying data.Considering the inherent scarcity of OR loss data the results of the statistical corre-lation analyses have to be carefully assessed. Qualitative methods including expertjudgement of reasonableness are important for validation. In particular, high (posi-tive or negative) correlations have to be reviewed in order to determine whether theyare justified or simply due to the lack of data.

Dependencies within a cell18To complicate matters further, many operational events do not have a clearly defined date of

occurrence but consist of a chain of incidents and are spread over a period of time.

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If loss events in a particular cell do not occur independently the frequency distri-bution is not Poisson. In case of positive correlation it might be more appropriateto model the frequency distribution in this cell as Negative Binomial. We refer tosection 6 for a detailed discussion.

The standard assumption in the LDA model is that

• frequency and severity distributions in a cell, i.e. Nk and Sk, are independentand

• the severity samples Sk1, . . . , SkNkare independent and identically distributed.

For validating the first assumption we have calculated rank correlations of frequencyand severity time series. The independence of severity samples has been testedby calculating Durban-Watson autocorrelation coefficients of monthly severity dataand sequential losses respectively. In summary, these statistical analyses support theLDA standard assumptions stated above. It also seems questionable whether anyextension of the model is justified as long as there is not sufficient data available forits calibration. After all, the standard LDA model has been successfully applied inthe insurance industry for many years.

Dependencies between cellsIntuitively, dependencies within a cell are higher than dependencies between differentcells: all losses in a cell have the same event type and business line, they are oftenaffected by the same internal processes, control and management structures and haveoccurred in the same business environment. On the other hand, losses in differentcells, say fraud in retail banking and employee claims in investment banking, usuallyhave nothing in common. The regulatory debate, however, is rather focused onmodelling dependencies between cells, i.e. between different business lines or eventtypes (see, for example, CEBS(2006)).

Central issues for modelling dependencies between cells are:

• What is the best level for specifying dependencies: frequencies, severities orloss distributions?

• What are appropriate mathematical models for dependencies in operationalrisk?

In DB’s LDA model, dependencies of frequency distributions are specified. Thereare a couple of reasons for this choice.

• First of all, we think that frequencies are the natural concept for specifyingdependence: dependencies of loss events are usually reflected in the date ofoccurrence of these losses. As a consequence, the frequencies in the respectivecells will be positively correlated.

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• The statistical analysis of frequency correlations shows a positive average cor-relation although significant frequency correlations are found between specificsubsets of cells only.

• The third advantage is that the specification of dependencies at frequency leveldoes not complicate the simulation of loss distributions (see section 9).

We do not assume positive correlations between severity distributions. More pre-cisely, severity samples Ski, . . . , Slj are assumed independent for all i, j, k, l. Themain reasons are:

• We have calculated linear and rank correlations of monthly severity time seriesand have not observed severity correlations in our loss data except for lossesthat are caused by the same event. These so-called split losses are not modelledseparately but as one event with loss amount specified as the sum of the splitloss components (see subsection 3.3).

• From a conceptual point of view, modelling severity correlations is more diffi-cult than modelling frequency correlations (see Frachot et al., 2004): assumethat the severity distributions in cells k and l are correlated, i.e. Ski, Slj have apositive correlation for all i = 1, . . . , Nk, j = 1, . . . , Nl. It immediately followsthat also the severity samples within these cells are correlated, i.e. Ski, Slj havea positive correlation for all i = 1, . . . , Nk, j = 1, . . . , Nl. This is a contradictionto one of the underlying assumptions in the standard LDA.

Another approach to modelling severity correlations is based on a weaker as-sumption that is consistent with the standard LDA model. It is used in com-mon shock models (Lindskog and McNeil, 2001): instead of correlating severitysamples Ski in cell k to all samples Sl1, . . . , SlNl

in cell l it is only assumed thatSki is correlated to one specific sample Slj . Hence, there exists a link betweenSki and Slj , for example both losses are caused by a common shock. Since com-mon shock events correspond to split losses in the internal data set this “weak”form of severity correlation is already incorporated in DB’s LDA model.

Some authors propose to model dependencies between the aggregate losses Lk (seeDi Clemente and Romano (2004), Chapelle et al. (2004), Nystrom and Skoglund(2002)). We think, however, that it is more intuitive to directly induce dependen-cies between the components of the aggregate losses, i.e. frequencies or severities.Moreover, the dependence of aggregate losses has to be applied in the context of cap-ital calculation after generation of the aggregate loss distribution and incorporationof insurance mitigation. As the insurance model adds another layer of dependencebetween cells it would be particularly hard to find a meaningful way to apply thisapproach.

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7.2 Modelling frequency correlations

We model dependencies of frequency distributions with copula functions. Copulasare the most general mathematical concept for specifying dependence (see Nelson(1999) for an introduction). Their application to operational risk has been proposedin a number of papers, for instance in Ceske et al. (2000), Song (2000), Frachot et al.(2001), Frachot et al. (2004) and Bee (2005). For alternative mathematical conceptslike common shock models and factor models we refer to Lindskog and McNeil (2001),Kuhn and Neu (2002), Nystrom and Skoglund (2002) and Powojowski et al. (2002).

In the current data environment, the selection of a copula family for modellingfrequency correlations cannot be based on statistical tests. For simplicity reasons,we have decided to model dependencies of frequency distributions with Gaussiancopulas. The correlation estimates are chosen very conservatively in order to takeinto account the uncertainty regarding the correct copula family.

Let m be the number of cells, F1, . . . , Fm the distribution functions of the fre-quency distributions in the individual cells and CΣ the Gaussian copula with corre-lation matrix Σ. Then the distribution function

CΣ(F1(x1), . . . , Fm(xm))

specifies the m-variate frequency distribution.It remains to specify the correlation matrix of the Gaussian copula. In princi-

ple, this matrix can be derived in different ways. There are two main approaches:multivariate and bivariate estimators. The bivariate estimators determine the corre-lation between two cells and use the results to construct the multivariate correlationmatrix. Motivated by simulation results in Lindskog (2000) we apply a bivariatemethod.

Even though the Gaussian copula is specified by a linear correlation matrix (Pear-sons Correlation Coefficient), we use a nonparametric rank estimator. The key con-cept of a rank estimator is to rank the observed data and derive the correlationcoefficients from these ranks. The main advantage over the common sample (lin-ear) correlation estimator lies in the fact that the distribution of the ranks is exactlyknown, namely a uniform distribution. Of course, some information is lost by replac-ing the observed values by their ranks, but the advantages over the standard linearestimator are significant. In particular, rank estimators are not sensitive to outliersand they are independent of the underlying distributions (compare to Embrechts etal. (1999)).

Having aggregated the frequency data on a monthly basis, we estimate the cor-relation between the cells by Spearman’s Rho. This rank estimator is the commonsample correlation estimator applied to the ranks of the data. See McNeil et al.(2005) and Chapelle et. al (2004) for its application and test results.

Our analysis indicates that significant frequency correlations are found betweenspecific subsets of cells only: the average correlation is around 10%, higher correla-

31

tions are only found between cells of the event type Execution, Delivery and ProcessManagement. In the EC calculations in 2005 and 2006, a homogeneous correlationmatrix with a correlation coefficient of 50% is used. In spite of this very conservativesetting, the correlation between the aggregated loss distributions in the individualcells is rather low (due to independent severity distributions). As a consequence, thecapital estimates are insensitive to frequency correlations or copula assumptions forfrequencies. (We refer to Bee (2005) for a comparison between Gaussian and Gum-bel copulas applied to frequencies and their impact on capital estimates.) In otherwords, we have experienced strong diversification effects, i.e. the 99.98%-quantile ofthe loss distribution at Group level is significantly lower than the sum of the quan-tiles of the loss distributions in the cells. This feature of the model is in line withthe results reported in Frachot et al. (2004). Chapelle et al. (2004) also find lowor even negative correlations between frequencies and observe a large diversificationbenefit.

8 Risk mitigation

8.1 Insurance models in the Loss Distribution Approach

Insurance is a well-established risk management tool that has been used by thebanking sector for many decades. Critical to the viability of recognizing insuranceas a risk mitigant in the Basel II Accord is a sound methodology for calculating thecapital relief resulting from insurance.19 A number of approaches based on insurancepremia, limits or actual exposures have been proposed (see, for instance, InsuranceCompanies (2001)). The general set-up of DB’s insurance model is tailored to theLoss Distribution Approach and follows the principles laid out in the working paperby the Insurance Companies (2001). Similar models are discussed in Brandts (2005)and Bazzarello et al. (2006).

Compared to other methodologies the Loss Distribution Approach has the ben-efit of allowing a fairly accurate replication of the risk profile of a bank, includingthe risk reducing effect of insurance. Transferring risk to an insurer through in-surance products alters the aggregate loss distribution by reducing the severity oflosses that exceed the policy deductible amount. The frequency of loss is unaffectedby insurance. More precisely, when frequency and severity distributions are com-bined through simulation, each individual loss point can be compared to the specificinsurance policies purchased by the bank and the corresponding policy limits and de-ductibles. As a consequence, an insurance model in the Loss Distribution Approachconsists of two main components:

• a quantitative model of the individual insurance contracts,19The recognition of insurance mitigation will be limited to 20% of the total operational risk

capital charge for regulatory purposes.

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• a mapping from the OR event types to the insurance contracts.

8.2 Modelling insurance contracts

Insurance contracts are characterized by certain specifications regarding the com-pensation of losses:

• A deductible d is defined to be the amount the bank has to cover by itself.

• The single limit l of an insurance policy determines the maximum amount of asingle loss that is compensated by the insurer. In addition, there usually existsan aggregate limit lagg.

• The recognition of the risk mitigating effects of insurance contracts within theAMA is subject to a number of prerequisites by the regulators. In particu-lar, the methodology for recognizing insurance has to take into account thecertainty of payment, the speed of payment, insurance through captives or af-filiates and the rating of the insurer. We translate the rating into a defaultprobability p and model defaults of the insurer by the binary random variableIp, where Ip = 0 with probability p and Ip = 1 otherwise. The other limita-tions are transferred by expert judgement into a haircut H which is applied tothe insurance payout.

In mathematical terms, the amount paid by the insurer is defined by

Ip ·max(min(l, x)− d, 0) ·H.

The actual specification of insurance parameters including haircuts has to be basedon insurance data and expert knowledge and is beyond the scope of this paper.

8.3 Mapping OR event types to insurance policies

In the OR capital calculation, insurance contracts are applied to simulated OR losses.This requires a mapping from OR loss categories, e.g. OR event types, business linesor a combination of both, to insurance contracts. If all losses of one OR category arecovered by the same insurance policy, this mapping is trivial. If the losses of one ORcategory can be covered by several insurance policies, the percentage of losses thatfall into a specific insurance category need to be determined by expert assessment andhistorical data. For example, 70% of execution losses might be covered by GeneralLiability, 20% by Professional Liability and 10% are not insured. A mapping fromOR Level 3 Loss Events to standard insurance products can be found in InsuranceCompanies (2001).

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9 Calculation of Economic Capital and capital alloca-tion

In the previous sections the quantitative components of the loss distribution modelhave been presented. This section focuses on the simulation of the aggregate lossdistribution (including insurance) at Group level and the calculation and allocationof Economic Capital.

9.1 Risk measures and allocation techniques

In recent years, a well-founded mathematical theory has been developed for measur-ing and allocating risk capital. The most important cornerstone is the formalizationof the properties of a risk measure given in Artzner et al. (1999). Their axiom-atization provides an appropriate framework for the analysis and development ofrisk measures, e.g. Expected Shortfall (Rockafellar and Uryasev, 2000; Acerbi andTasche, 2002). General principles for capital allocation can be found in a number ofpapers, for example in Kalkbrener (2005).

The general theory of measuring and allocating risk is independent of specific risktypes. In fact, it is important that the same measurement and allocation techniquesare applied in the Economic Capital calculation for market, credit and operationalrisk. This is a prerequisite for a consistent framework for measuring risk and perfor-mance across the bank.

Economic Capital is a measure designed to state with a high degree of certaintythe amount of capital needed to absorb unexpected losses. The exact meaning of“high degree of certainty” depends on the risk tolerance of the bank. A popular ruleis to choose the tolerance level that reflects the average default rate associated withthe rating of the bank. This rule immediately translates into a definition of EconomicCapital based on Value-at-Risk (defined as a quantile of the loss distribution): forexample, Deutsche Bank’s EC is determined by the Value-at-Risk at confidence level99.98% reflecting the AA+ target rating of the bank. The simple link between ratingsand Value-at-Risk based Economic Capital is one reason why this specific EconomicCapital methodology has become so popular and has even achieved the high statusof being written into industry regulations.

The confidence level of 99.98% requires a high precision in modelling tails ofloss distributions. It is obvious that this problem is particularly challenging foroperational risk. In fact, this very high confidence level is one of the reasons whyinternal data has to be supplemented by external losses. Even if external lossesand generated scenarios are used as additional information the 99.98% quantile (andto a lesser degree the 99.9% quantile) has to be based on extrapolation techniquesbeyond the highest losses in the data base. It is inevitable that there is some degreeof uncertainty about the correct value of these high quantiles due to the scarcity of

34

data in the extreme tail.20 We refer to Mignola and Ugoccioni (2006) and to section11 for more information.

The standard procedure for credit and operational risk is to specify the EconomicCapital as Value-at-Risk minus Expected Loss. In credit risk, Expected Loss denotesthe mean of the portfolio loss distribution. If the same definition is applied in opera-tional risk the Expected Loss significantly exceeds the annual loss in an average year:due to the fat-tailed severities the mean of the aggregate OR loss distribution is usu-ally much higher than its median. This is the reason why alternative definitions ofExpected Loss are analyzed, e.g. replacing the mean of the aggregate loss distribu-tion by the median (see Kaiser and Kriele (2006) for approaches to the calculation ofEL estimates). In Deutsche Bank, expert opinion is used to transform an initial ELestimate based on historic internal loss data into an EL forecast for the one-year timehorizon. Using raw internal losses as starting point for an EL estimate has the ad-vantage that the Expected Loss methodology is based on internal data only and thatno sophisticated statistical techniques are required. Furthermore, the involvement ofOR experts from the different divisions of the bank ensures that the quantificationof the Expected Loss is closely linked to the budgeting and capital planning processof the bank. This feature of the EL quantification process is particularly importantfor meeting the regulatory requirement that the bank has measured and accountedfor its EL exposure. This requirement is a precondition for subtracting EL from the99.9% Value-at-Risk in the calculation of Regulatory Capital.

The allocation of risk capital to BL/ET cells is based on Expected Shortfallcontributions. Intuitively, the Expected Shortfall contribution of a BL/ET cell isdefined as its contribution to the tail of the aggregate loss distribution:

E(Lk |L > VaRα(L)),

where Lk denotes the loss distribution in cell k, L is the aggregate loss distributionat Group level and α has been chosen such that

E(L |L > VaRα(L)) = VaR0.9998(L).

Expected Shortfall allocation has a number of theoretical and practical advantagescompared to classical allocation techniques like volatility contributions (Kalkbreneret al., 2004). Furthermore, it is consistent with the methodology used in DB’s ECmethodology for credit risk and risk type diversification.

In the following two subsections, the Monte Carlo based EC calculation andallocation is described in more detail.

20An alternative approach would be the replacement of the direct calculation of the 99.98% quan-tile by the calculation of a lower quantile and a subsequent scaling. However, the scaling factor forthe 99.98% quantile is as uncertain as the direct calculation of the 99.98% quantile and thereforedoes not improve the reliability of the EC estimate.

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9.2 Simulation of the aggregate loss distribution

The aggregate loss distribution at Group level cannot be represented in analyticform. In order to overcome this problem, different approximation techniques havebeen proposed in the literature including Panjer recursion, fast Fourier transformmethods and approximations based on properties of subexponential distributions(Klugman et al., 2004; McNeil et al., 2005; Bocker and Kluppelberg, 2005). However,all these techniques limit the functionality of the underlying model. In particular,it becomes more difficult to incorporate advanced modelling features like insuranceor more sophisticated allocation techniques. We have therefore chosen Monte Carlosimulation for calculating and allocating Economic Capital. This decision has beenmade despite the fact that Monte Carlo estimates of high quantiles of the aggregateloss distribution and therefore MC estimates of EC are numerically unstable. Inorder to reduce the number of required MC samples we have developed a semi-analytic technique for Expected Shortfall allocation. Furthermore, we are workingon the adaptation of importance sampling techniques to LDA models. Importancesampling has been successfully applied in market and credit risk (see Glassermannet al. (1999), Kalkbrener et al. (2004) and Glasserman (2003)). The adaptationof this variance reduction technique to operational risk requires the introductionof rare event simulation techniques for heavy-tailed distributions (see, for example,Asmussen et al., 2000). We will report on the variance reduction techniques used ina forthcoming paper.

These are the main steps in generating one MC sample of the aggregate lossdistribution at Group level:

Simulation of loss distributions in individual cells

• Generate a sample (n1, . . . , nm) of the m-dimensional frequency distributionspecified by the Gaussian copula CΣ. This can easily be done by a standardsoftware package or by implementing a simulation algorithm as proposed ine.g. McNeil et al. (2005).

• Compute nk samples sk1, . . . , sk nKof the severity distribution in cell k, k =

1, . . . ,m.

Incorporation of insurance

• Take all losses in all cells and order them randomly.

• Determine the insurance policy (if available) for each loss.

• Apply the insurance model to the ordered loss vector and compute the corre-sponding vector of net losses (loss minus compensation).

Sample of aggregate loss distribution

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• The sum of the net losses is a sample l of the aggregate loss distribution atGroup level.

9.3 Monte Carlo estimates for Economic Capital

The simulation of the aggregate loss distribution is used as input for the calculationand allocation of Economic Capital: assume that an ordered list l1 > . . . > ln ofsamples of the aggregate loss distribution at Group level has been computed. TheEC computation is done in several steps:

EC calculationThe Economic Capital (before qualitative adjustment) is calculated as the 99.98%quantile of the aggregate loss distribution at Group level minus the Expected Loss.Note that the 99.98% quantile is represented by lq, where q := n/5000.

EC allocation to BL/ET cellsThe Economic Capital is then allocated to the cells of the BL/ET matrix using Ex-pected Shortfall techniques. More precisely, define an index r and consider the rhighest losses l1 > . . . > lr as a representation of the tail. In DB’s LDA model, r ischosen such that the sum

∑ri=1 li/r is approximately the 99.98% quantile lq . Note

that it is easy to compute a decomposition li =∑

lik, where lik denotes the contri-bution of the cell k to loss li: since each loss li is a sum of losses in individual cellsthe lik are obtained as byproducts in the simulation process. The capital allocatedto cell k is the Expected Shortfall contribution defined by

∑ri=1 lik/r.

EC allocation to business linesThe EC of the individual business lines (before qualitative adjustment) is obtainedby aggregation of the Economic Capital allocated to the BL/ET cells: the capital ofeach business line is the sum of

• the capital allocated to its three cells covering the event types Fraud andClients, Products, Business Practices and Execution, Delivery, Process Man-agement and

• a proportional contribution from the Group cells and the cells for the eventtypes Infrastructure and Employment Practices, Workplace Safety.

The above procedure ensures that the entire amount of EC for the Group is allocatedto the business lines.

The sub-allocation of EC to business units within business lines cannot be basedon Expected Shortfall techniques since separate loss distributions are not specifiedfor these units. Deutsche Bank applies risk indicators instead.

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Qualitative adjustment of ECThe EC of each business line is scaled by a multiplicative factor that reflects qualita-tive adjustments. The Economic Capital at Group level has to be modified accord-ingly in order to ensure additivity of EC figures, i.e. the adjusted EC of the Groupis obtained by adding up the adjusted EC figures of the individual business lines.More information on qualitative adjustments is given in the following section.

10 Incorporation of business environment and internalcontrol factors

Apart from generated loss scenarios LDA models mainly rely on loss data and areinherently backward looking. It is therefore important to incorporate a componentthat reflects changes in the business and control environment in a timely manner.

There have been intense discussions in the industry regarding different strategiesa bank could follow in order to perform qualitative adjustments. The ways that ad-justments are performed vary in form and complexity and the processes for collectingrelevant information differ across financial institutions. Common industry practiceis to compile the data into a scoring mechanism which translates qualitative infor-mation into numerical values. The most prevalent forms of qualitative adjustmentutilize data from key risk indicators (KRIs) and risk self-assessment.

We will not discuss these general topics in this paper but focus on the incorpora-tion of qualitative adjustments into LDA models.21 Qualitative adjustments mightoccur in different components of an LDA model, e.g. adjustments of

• the parameters of frequency distributions,

• the parameters of severity distributions, or

• the Economic Capital of business lines.

We have decided to directly adjust the allocated Economic Capital of a business line(third option). The main reasons for this decision are simplicity and transparency:

• The application of qualitative adjustments to the Economic Capital separatesthe quantitative EC calculation from the qualitative adjustments and thereforereduces the complexity of the model.

• It is important for the acceptance of the methodology by the business thatthe impact of qualitative adjustments on allocated Economic Capital is astransparent as possible. This transparency is only guaranteed by applyingadjustments directly to the allocated EC.

21For more information on self-assessment and KRIs we refer to Anders and Sandstedt (2003) andDavies et al. (2006)

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• A further consequence of a direct EC adjustment is that QA for one businessunit does not affect the other business units.

The direct application of qualitative adjustments to Economic Capital is difficult tojustify with statistical means. However, we think that the advantages listed aboveoutweigh this disadvantage at the current stage of the model development. We alsobelieve that the statistical validation of QA techniques is one of the main challengesfor the development of the next generation of LDA models.

11 Analysis and validation of LDA models

The final section of this paper deals with model analysis and validation. The tech-niques and results in this section are not restricted to Deutsche Bank’s LDA modelbut can be applied to a more general class of models (see section 11.1.1 for a defini-tion).

Basel II requires that an operational risk AMA model generates a capital measureto the 99.9%-ile confidence level over a one-year holding period. The EconomicCapital of Deutsche Bank is derived from an even higher quantile, i.e. the 99.98%quantile of the aggregate loss distribution for one year. The objectives of modelvalidation are to show that the estimates for Regulatory and Economic Capital arereasonable. Given these high quantiles it is obvious that a validation of OR capitalrequirements is a very ambitious task. But not only demanding confidence levelscomplicate the application of statistical validation techniques in operational risk,in particular compared to market risk: there are a number of additional problemsincluding

1. the shortage of relevant operational risk data,

2. the context dependent nature of operational risk data, and

3. the current lack of a strongly risk sensitive exposure measure in operationalrisk modelling (cf market and credit risk).

As a consequence, operational risk measurement and statistical validation techniquesinevitably suffer from a higher degree of uncertainty than market risk validationand have to be combined with qualitative methods including expert judgement ofreasonableness.

The validation of an operational risk model involves a review of data inputs,model methodology (including any assumptions contained within the model suchas correlations), and model outputs (AMA Working Group, 2004). Since the spe-cific methodology implemented in Deutsche Bank’s LDA model has been extensivelydiscussed in previous sections we now focus on sensitivity analysis and output val-idation. More precisely, we present a sensitivity analysis of the model components

39

frequencies, severities, dependence structure and insurance. This analysis comprisesthe assessment of

1. the capital impact of different modelling assumptions and

2. the sensitivity of capital requirements to parameter changes

for each model component (section 11.2). The analysis uses basic properties ofLDA models presented in sections 11.1.2 and 11.1.3 and is therefore not limitedto the model implemented at Deutsche Bank. Section 11.3 deals with the impactanalysis of stress scenarios. Section 11.4 briefly outlines the inherent problems withthe application of backtesting techniques to OR models and reviews some of thebenchmarks for AMA capital charges suggested in the literature. However, the mainfocus of section 11.4 is on an approach for benchmarking quantiles in the tail of theaggregate loss distribution of the LDA model against individual data points fromthe underlying set of internal and external losses.

11.1 Model characteristics

11.1.1 Definition of a general class of LDA models

The formal framework for our analysis is a model which is based on m distinct (ag-gregate) loss distributions. These loss distributions may correspond to combinationsof business lines and event types, i.e. cells in the BL/ET matrix. The specificationof each loss distribution follows an actuarial approach: separate distributions forloss frequency and severity are derived and then combined to produce the aggregateloss distribution. More precisely, let N1, . . . , Nm be random variables which repre-sent the loss frequencies, i.e. the values of Ni are non-negative integers. For eachi ∈ {1, . . . , m}, let Si1, Si2, . . . be iid random variables independent of Ni. Thesevariables are independent samples of the severity distribution Si of loss type i. Theloss variable Li in this cell is defined by

Li :=Ni∑r=1

Sir.

The aggregate loss distribution at Group level is defined by

L :=m∑

i=1

Li.

11.1.2 Variance analysis

Variance analysis does not provide information on quantiles of loss distributions but

40

1. it quantifies the impact of frequencies, severities and their dependence structureon the volatility of aggregate losses and

2. it is independent of specific distribution assumptions, i.e. the first two momentsof the aggregate loss distribution only depend on the first two moments of thefrequency and severity distributions.

Variance analysis therefore is a rather general technique that provides insight intothe significance of the individual model components.

For calculating moments we make the additional assumption that severity sam-ples in different cells are independent, i.e. Sik and Sjl are independent for i, j ∈{1, . . . , m} and k, l ∈ N. However, we allow for dependence between the frequencyvariables N1, . . . , Nm.

Under the assumption that the first and second moments of frequency distribu-tions Ni and severity distributions Si exist we obtain the following formulae for theexpectation and the variance of the aggregate loss distribution at cell and Grouplevel:

E(Li) = E(Ni) · E(Si) (2)

E(L) =m∑

i=1

E(Ni) · E(Si) (3)

Var(Li) = E(Ni)Var(Si) + Var(Ni)E(Si)2 (4)

Var(L) =m∑

i=1

E(Ni)Var(Si) + Var(Ni)E(Si)2 +

m∑i,j=1, i 6=j

Cov(Ni, Nj)E(Si)E(Sj) (5)

The derivation of these formulae is a rather straightforward generalization of theproof of Proposition 10.7 in McNeil et al. (2005)).

Formula (5) provides the general representation of Var(L) for the model classspecified in section 11.1.1. In order to better understand the impact of the differentmodel components, i.e. frequencies, severities and frequency correlations, it is usefulto simplify formula (5) in an idealistic homogeneous model. More precisely, weassume that

1. frequencies in all cells follow the same distribution N ,

2. severities in all cells follow the same distribution S, and

3. all frequency correlations equal a fixed c, i.e. for i, j ∈ {1, . . . , m}

c = Corr(Ni, Nj) = Cov(Ni, Nj)/Var(N).

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Then the variance Var(L) of the aggregate loss distribution can be written in theform

Var(L) = m · (E(N) ·Var(S) + Var(N) · E(S)2 · (c · (m− 1) + 1)). (6)

11.1.3 Loss distributions for heavy-tailed severities

Since OR capital requirements of large international banks are mainly driven by rareand extreme losses it is quite natural to work with severity distributions that havebeen applied in insurance theory to model large claims. These distributions typicallybelong to the class of subexponential distributions.

The defining property of subexponential distributions is that the tail of the sumof n subexponential random variables has the same order of magnitude as the tail ofthe maximum variable. More precisely, let (Xk)k∈N be positive iid random variableswith distribution function F and support (0,∞). Then F is subexponential if for alln ≥ 2,

limx→∞

P(X1 + . . . + Xn > x)P(max(X1, . . . , Xn) > x)

= 1. (7)

The following property justifies the name subexponential for distributions satis-fying (7): if the distribution function F is subexponential then for all ε > 0,

eεxF (x) →∞, x →∞

where F (x) = 1− F (x) denotes the tail of F . Hence, F (x) decays to 0 slower thanany exponential eεx. Examples of subexponential distributions include the familiesPareto, Weibull (τ < 1), lognormal, Benktander-type-I and -II, Burr, loggamma.

Under certain conditions, random sums of subexponential variables are subex-ponential. Let (Xk)k∈N be iid subexponential random variables with distributionfunction F . Let N be a random variable with values in N0 and define

p(n) := P(N = n), n ∈ N0.

Denote the distribution function of the random sum∑N

i=1 Xi by G, i.e.

G(x) = P(N∑

i=1

Xi ≤ x), x ≥ 0.

Suppose that p(n) satisfies

∞∑n=0

(1 + ε)np(n) < ∞ (8)

42

for some ε > 0. Then

the distribution function G is subexponential and limx→∞

G(x)F (x)

= E(N). (9)

If N follows a Poisson or negative binomial distribution then (8) is satisfied (seeexamples 1.3.10 and 1.3.11 in Embrechts et al. (1997)).

The above result states that the tail of the aggregate loss distribution only de-pends on the tail of the subexponential severity distribution. An explicit approxi-mation formula for G(x) can be found in Bocker and Kluppelberg (2005).

For the total sum L :=∑m

i=1 Li, results similar to (9) hold under extra conditions.We refer to Embrechts et al. (1997) and McNeil et al. (2005).

11.2 Sensitivity analysis of LDA models

In the previous subsection, basic facts on frequencies, severities and aggregate lossdistributions have been presented. These results will now be used to analyze for eachmodel component:

1. the capital impact of different modelling assumptions and

2. the sensitivity of capital requirements to parameter changes.

Sensitivity analysis gives an impression about which inputs have a significant impactupon the risk estimates and by implication, need prioritizing in terms of ensuringtheir reasonableness.

11.2.1 Frequencies

The impact of the shape of frequency distributions on capital requirements is ratherlimited. The reason is that OR capital is mainly driven by cells with fat-tailedseverities. In those cells, the distribution assumptions for frequencies have no signif-icant impact on the aggregate loss distributions. This is a consequence of (9): forsubexponential severities, the tail of the aggregate loss distribution is determined bythe tail of the severity distribution and the expected frequency (but not its preciseshape). This result is in line with formula (4) which says that the impact of the fre-quency distribution on the variance of the aggregate loss distribution depends on therelationship of the frequency vol Var(N)/E(N) and the severity vol Var(S)/E(S)2.In high impact cells, the volatility of severities dominates and the actual form ofthe frequency distribution is of minor importance. These theoretical results are con-firmed by our own test calculations discussed in section 6 and by a number of papersincluding Bocker and Kluppelberg (2005) and De Koker (2006).

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11.2.2 Severities

OR capital requirements are mainly driven by individual high losses. Severity dis-tributions specify the loss size and are therefore the most important component inquantitative OR models. This claim is again supported by formula (4) and the resultin (9): in a cell with fat-tailed severity distribution the volatility and the shape ofthe aggregate loss distribution are mainly determined by the severity distribution.All relevant aspects of severity models therefore have an impact on the aggregateloss distribution and on capital estimates. In particular, OR capital requirementsdepend on

1. weights and techniques for combining different data sources for severity mod-elling and

2. distribution assumptions for severities, in particular for severity tails.

11.2.3 Insurance

The impact of insurance on risk capital depends on the insurance coverage of thedominant OR event types. In Deutsche Bank’s LDA model, most of the OR capitalis allocated to the event type Clients, Products and Business Practices. The highlosses in this event type are predominantly classified as insurance type ProfessionalLiability (PL). Since individual extreme losses are the main driver for capital thesingle limit for Professional Liability is a very significant insurance parameter inthe current model setup. Another important parameter is the actual coverage ofProfessional Liability, i.e. the proportions between insured and uninsured PL losses.

11.2.4 Dependence

In a bottom-up Loss Distribution Approach, separate distributions for loss frequencyand severity are modelled at ”business line/event type” level and are then combined.In this basic LDA model, dependencies between the occurrence of loss events andthe size of losses might occur in a cell or between different cells.

If loss events in a particular cell do not occur independently the frequency distri-bution is not Poisson. In case of positive correlation it might be more appropriateto model the frequency distribution in this cell as Negative Binomial. The impact ofthese distribution assumptions for frequencies has already been discussed in section11.2.1.

The impact of dependencies between cells depends on the level where these de-pendencies are modelled, e.g. frequencies, severities or aggregate losses. Equality (6)provides information on the impact of frequency correlations on capital requirements:it depends on

1. the number of (relevant) cells m and

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2. the relationship of V ar(F )/E(F ) (frequency vol) and V ar(S)/E(S)2 (severityvol).

In Deutsche Bank’s LDA model, for example, the number of relevant (= high impact)cells is relatively small and V ar(S)/E(S)2 clearly dominates V ar(F )/E(F ) in thesecells. The impact of frequency dependence is therefore insignificant. In contrast, de-pendence between severity distributions or loss distributions has a significant impacton capital. For example, assume that loss distributions in different cells are comono-tonic. Then every α-quantile of the aggregate loss distribution (Group level) equalsthe sum of the α-quantiles of the loss distributions in the cells (see, for example,McNeil et al. (2005)). For α = 0.9998, this sum would clearly exceed the EconomicCapital calculated in Deutsche Bank’s LDA model.

11.3 Impact analysis of stress scenarios

Stress testing has been adopted as a generic term describing various techniques usedby financial firms to gauge their potential vulnerability to exceptional but plausibleevents. In a typical OR stress scenario, internal and/or external losses are added orremoved and the impact on capital is analyzed. The scenarios are often provided bybusiness and OR management to quantify

1. potential future risks,

2. the impact of business strategies, and

3. risk reduction by OR management.

In this context, stress scenarios have a wider scope than scenario analysis which islimited to the generation of single loss data points (compare section 3.4.3).

Stress tests also play an important role in model analysis and validation. Theobjective is to study the response of the model to changes in model assumptions,model parameters and input data. In particular, we have analyzed the sensitivity ofthe model to changes in the loss data set. These stress tests are an important partof the model development.

A different application of stress scenarios is discussed in the next section: LDAmodel outputs are tested by comparison with adverse extreme, but realistic, scenar-ios.

11.4 Backtesting and benchmarking

Backtesting, as the name suggests, is the sequential testing of a model against re-ality to check the accuracy of the predictions. More precisely, the model outcomesare compared with the actual results during a certain period. Backtesting is fre-quently used for the validation of market risk models. In credit and operational risk,

45

the inherent shortage of loss data severely restricts the application of backtestingtechniques to capital models.

The most useful validation of an LDA model would be the backtesting of cap-ital estimates against actual annual losses. However, it is obvious that capital re-quirements derived from high quantiles of annual loss distributions cannot be testedagainst the actual loss history of a single bank in a reasonable way.22 In this paperwe propose a different approach. It is based on the assessment that OR capitalrequirements are mainly driven by individual high losses. We argue that individualdata points from the set of internal and external losses can therefore be used asbenchmarks for quantiles in the tail of the aggregate loss distribution of the LDAmodel. More precisely, the basic idea is to compare

1. the tail of the aggregate loss distribution calculated in a bottom-up LDA model,e.g. Deutsche Bank’s LDA model, and

2. the tail of an aggregate loss distribution directly specified at Group level.

An approximation of the tail specified at Group level can be directly derived fromthe loss data, i.e. each quantile of the aggregate loss distribution is represented by asingle data point. Hence, this method allows to compare quantiles of the aggregateloss distribution in a bottom-up LDA model to individual losses from the underlyingset of internal and external loss data.

We will now describe the proposed technique in more detail. The first step is thespecification of an aggregate loss distribution at Group level, i.e. all internal andexternal losses across business lines and event types are considered. However, sincewe are only interested in the tail of the distribution we take only losses above a highthreshold, say 1m, into account. Let n denote the bank’s average annual loss fre-quency above 1m. We assume that the frequency distribution satisfies the conditionin (8) and the severity distribution S at Group level is subexponential. Let S1, . . . , Sn

be independent copies of S. It follows from (7) and (9) that the α-quantiles of theaggregate loss distribution and the maximum distribution max(S1, . . . , Sn) convergefor α → 1. The α-quantile of the maximum distribution is now identified with the1 − ((1 − α)/n)-quantile of the severity distribution. The latter is directly derivedfrom the data set for appropriate α’s. Note that the amount of loss data providesa limit for the confidence level α, i.e. depending on the annual loss frequency n the1− ((1−α)/n)-quantile of the empirical severity distribution is only specified up toa certain α.

In summary, we propose to compare the α-quantile of the aggregate loss distri-bution calculated in an LDA model and the 1− ((1− α)/n)-quantile of the empiric

22It should be noted that this situation will not improve over time, since context dependencyof OR loss data will mean that 5-10 years of data will probably be the maximum useful data setagainst which banks can validate their capital models (depending on the risk type and the speed ofchange in environment and control processes).

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severity distribution derived from the complete set of internal and relevant externallosses above 1m. Application of this method to DB’s LDA model yields the resultsshown in figure 4: the tail of the aggregate loss distribution has the same magnitudeas the corresponding high impact losses in the underlying data set.

Figure 4: 1y aggregate loss distribution from AMA model vs. corresponding highimpact losses in the underlying data set.

Our approach has a lot in common with the methodology proposed in De Koker(2006). First of all, both approaches are based on the assumption that the size of asingle (super-sized) loss event provides a meaningful capital estimate. The likelihoodof these events is derived from the frequency above a high threshold and the shapeof the severity distribution. The main difference between both approaches is thatDe Koker (2006) approximates the severity distribution above the threshold by apower law whereas we use an empirical distribution, i.e. raw loss data. Empiricaldistributions have the advantage that no assumptions on the shape of the severitydistribution have to be made but, in contrast to the technique in De Koker (2006),only quantiles up to a certain confidence level can be benchmarked.

Other benchmarks for OR risk estimates have been proposed in the literature:

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1. comparison of a bank’s operational risk capital charge against a bank’s closepeers,

2. comparison of the AMA capital charge against the BIA or TSA capital charges,

3. comparison of the LDA model outputs against adverse extreme, but realistic,scenarios.

Although these tests help to provide assurance over the appropriateness of the levelof capital there are obvious limitations. Peer comparison will not indicate whetherbanks are calculating the right level of risk capital, just whether they are consistentwith each other. Given that the TSA/BIA are non-risk sensitive methodologies, andthe AMA is risk-sensitive, the value of any comparison between AMA and TSA/BIAcapital charges is questionable. Validation through scenarios is based on the iden-tification of an appropriate range of specific OR events and the estimation of lossamounts in each of these scenarios. This process is highly subjective and requiressubstantial resources for specifying a comprehensive operational risk profile of abank.

Given the limitations of OR validation techniques it is particularly importantthat a model meets the “Use Test”, i.e. it is incorporated within the bank’s oper-ational risk management framework and used to manage the risks facing the bank.Although the Use Test does not validate the capital estimates calculated in the modelit provides assurance that the model is working appropriately.

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